Step by Step Solutions - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim

Home : Support : Online Help : System : Information : Updates : Maple 2022 : Step by Step Solutions

Step-by-Step Solutions


Simplification Steps

Steps for Sketching a Curve

Simplification Steps

Maple 2022 includes new commands for showing the steps needed to manipulate algebraic expressions in order to reduce them to their simplest form.

In general the generated steps try to find that hard balance between being too verbose and too cryptic.  The SimplifySteps command errs on the side of adding more steps, and is aimed to help someone who wants to learn what the steps are, even for fundamentals like adding fractions.  Depending on the problem it will adjust somewhat; recognizing higher level problems for which it will decide to skip more easy-level steps.

Here are some examples of different categories of problems that SimplifySteps as well as related command FractionSteps can handle:





There is a dedicated FractionSteps command that goes into slightly more detail than SimplifySteps

FractionSteps 1/2 + 1/6

Let's Simplify Fractions12+16Find fractions to get lowest common denominator of63312+1116Multiply316+116Add numerators31+116Multiply313+116Multiply113+16Add3+146Cancel out factor of223



SimplifySteps 1/2 + 1/6

Let's simplify12+16Find fractions to get lowest common denominator of63312+1116Multiply316+116Add fractions23




SimplifySteps sqrt(6)*sqrt(10)/sqrt(12) 

Let's simplify61012Pull out a factor of4=2from1261023Multiply in order to rationalize the denominator3361023Multiply the denominator36106Factor roots323256Combine22,333256Multiply325656Cancel out factor of65



Let's simplify6101213Multiply in order to rationalize the denominator122312236101213Multiply the denominator122361012Factor roots3232213232512Combine3233,21322233162213512Multiply33162213526521331612Multiply26521331612521331612Cancel out factor of125213316





Let's simplify2x3y33xy22Evaluate exponent3xy222x3y39x2y4Multiply2x3y39x2y418x5y7



Let's simplifyx3x5Apply the product ruleanam=an+mto add exponents with common basex3+5Add exponentsx8Solutionx8



Let's simplifybnmApply the integer power of a power rule,anm=anmbnm



Let's simplifyy5y4Cancel out factor ofy4providedy40y



Let's simplifyy−5y4Divide assumingy4≠ 01y9



Let's simplify12351234Apply the quotient rule:anam=anm1231Evaluate exponent1231123




SimplifySteps log[10](2)/log[10](5)+log[10](3) 

Let's simplifylog102log105+log103Use the log rule,logax=logbxlogbato express as a single logarithmlog52+log103Solutionlog52+log103


Note: This is different than how Maple's simplify command treats expressions like this, always converting to ln:






SimplifySteps %log10100+%log101000 

Let's simplifylog10100+log101000Evaluatelog101002+log101000Evaluatelog1010002+3Add2+35



Let's simplify5logzz4Apply the log rulelogamn=nlogam54logzzApply the log rulelogaa=154Multiply5420



You can also use the new PowerSteps command to get step by step results for problems with radicals, exponents, and logarithms.  


SimplifySteps 1+cot(x)^2 

Let's simplify1+cotx2ApplyPythagorastrig identity,cotx2=cscx211+cscx21ApplyReciprocal Functiontrig identity,cscx=1sinx1sinx2Evaluate1sinx2


SimplifySteps"sinPi + Pi + x"

Let's simplifysinπ+π+xAddπ+π+xsin2π+xEvaluatesin2π+xsinx



Let's simplifysecx21secx2ApplyReciprocal Functiontrig identity,secx=1cosxsecx211cosx2ApplyPythagorastrig identity,secx2=1+tanx21+tanx21cosx2ApplyQuotienttrig identity,tanx=sinxcosxsinxcosx2cosx2Evaluatesinx2



Let's simplify−1cosx2+11+cotx2ApplyPythagorastrig identity,cotx2=cscx21cosx2+11+cscx21ApplyReciprocal Functiontrig identity,cscx=1sinxcosx2+11sinx2ApplyPythagorastrig identity,cosx2=1sinx21sinx2+1sinx2Evaluate1



You can also use the new TrigSteps command to get these step by step results.  


Showing the steps to solving an integral, limit, or derivative has been available in past versions of Maple via the Student:-Calculus1:-ShowSolution command. You can now also access those step by step solutions through SimplifySteps, further unifying the ability to do step by step solutions using a single command.


Let's simplifyx2+xⅆxIntegralto evaluatexⅆx1. Apply thepowerrule to the termxⅆxRecall the definition of thepowerrule, for n-1xnⅆx=xn+1n+1This means:xⅆx=x1+11+1So,xⅆx=x22We can rewrite the integral as:x22Sub evaluatedintegralback in expression3x22



Steps for Sketching a Curve

Maple 2022 includes a new command for showing the steps needed to sketch the graph of an expression by identifying the basic function and the transformations done to the function.  Various kinds of expressions are handled, including trig, logs, and polynomials to pick just a few.  Here are some examples:



Let's plot2sin3x+π3+1Compared to the plot ofsinx, we have a vertical stretch by a factor of22sin3x+π9+1Then, we have a horizontal compression by a factor of132sin3x+π9+1Then, we have a vertical shift of12sin3x+π9+1Then, we have a horizontal shift ofπ92sin3x+π9+1Apply the horizontal shift and stretch to the range,x=2π..2π2π13+π9..2π13+π9=−2.443460953..1.745329252We can now plot using the information extractedPLOT...



Let's plot2x2+4x+102x2+4x+10Complete the square2x+12+8With the expression in vertex form we can extract valuable information2x+12+8The coefficient2of thex+12term indicates a parabola that opensupand has a verticalstretchof22x+12+8We have a horizontal shift of−1and a vertical shift of8which gives a vertex of (−1,8)2x+12+8We can now plot using the information extractedPLOT...



Let's plot4x+104x+10This is a line; find two points and draw a line through themy=4x+10Setx= 0 to solve for y intercepty=10This gives a y intercept of (0,10)y=10Set expresson to 0 to solve forxintercept0=4x+10Subtract4xfrom both sides04x=4x+104xSimplify−4x=10Divide both sides by−4−4x−4=10−4Simplifyx=52This gives anxintercept of (52,0)x=52By connecting through the two points we can plot the linePLOT...



Let's plot23x+2Rewrite the equation in the following form23x+23+0Compared to the plot of1x, we have a vertical stretch by a factor of2PLOT...Then, we have a horizontal compression by a factor of13PLOT...Then, we have a horizontal shift of23PLOT...The final plot with asymptotes in cyan aty=0andx=23isPLOT...



For more information, see the help page Student:-Basics:-CurveSketchSteps.